On projective simplicity of certain groups of rational points over algebraic number fields

It is proved that, if $G$ is a simply connected anisotropic absolutely simple algebraic group with rank $n\geqslant2$ defined over an algebraic number field and decomposable over a quadratic extension, then the group $G(K)$ of rational points is projectively simple, i.e. the factor group modulo the center is simple. Projective simplicity of algebraic groups of type $B_n$, $C_n$, $G_2$, $F_4$, $F_7$ is obtained as a corollary, and also the same for groups of type $E_8$ whenever the Hasse principle holds. In addition the problem of projective simplicity for groups of type $^{(1)}D_n$, $^{(2)}D_n$ ($n\geqslant4$) is reduced to the case of groups of type $A_3$.
Bibliography: 18 titles.